Unit of the Determinant of a 2-space Matrix

Do determinants of a 2-space matrix have a unit of [units]^2?
Also, does A_parallelogram = || u x v|| have a unit of [units]^2 too?
This has confused me as Area has a unit of [units]^ 2 but the examples from the Contemporary Linear Algebra (Anton and Busby) text book does not state any units at all.

The area of a parallelogram has units of "u2" where u is whatever unit is being used to measure area. That is a specific application of norm and does NOT say that norm of the cross product of a vector has any units at all! Are you assuming that there is some form of "measurement" so that the vectors in whatever vector space you are talking about have a specific unit? And that raises the question, are you thinking of some specific vector space?

The area of a parallelogram has units of "u2" where u is whatever unit is being used to measure area. That is a specific application of norm and does NOT say that norm of the cross product of a vector has any units at all! Are you assuming that there is some form of "measurement" so that the vectors in whatever vector space you are talking about have a specific unit? And that raises the question, are you thinking of some specific vector space?

I had to calculate the area of N.M and the determinant of the N matrix.
And I was just wondering since it's the area (found by Aparallelogram = || u x v||) do I just write [units]^2?
Also since I'm meant to find out that the determinant is the area of the parallelogram, should I write [units]^2 or leave it blank?

-- So there is an application of the norm of the cross product of two adjacent vectors in 2-space (Aparallelogram = || u x v||), in this situation should I put [units]^2 in the answer? I deduce that I'm not meant to know about det(N) being the area of N.M so I guess I won't write any units for that.